Ever wonder about division?
The number 10 can be divided by 1, 2, 5 and itself 10. 10 has four divisors. 11 can only be divided by 1 and 11, with 2 divisors making it a prime. 12 can be divided by 1, 2, 3, 4, 6 and 12. 12 has 6 divisors and is also the smallest number with 6 divisors.
It makes you wonder what the smallest number would be with … let's say … 23 divisors. With some programming skill, knowledge of sieving and factorization techniques and some computer time you can find out. It's 4.194.304.
Ever since I used old punch card computers at my University too discover things like this, I've played around with numbers which can be divided by lots of other numbers. They are called highly compound numbers and I like them. I've run scans looking for these numbers up to the billions.
You can see a table of the first 28th of them at the left. You can hardly say there is a clear logical progression ... In the graph to the left these numbers are marked in blue. Red marks the smallest number with x divisors. Green marks the so called Perfect Numbers. They are the sum of their own divisors.
To the right you can see the DIVISOR function that returns the number of divisors for a given numbers. What do you think? Is there a pattern there?
12, 60 and 360
Some of them are quite well known.12 is the smallest with 6 divisors and this makes us split the 12 hours of the day efficiently. 60 is the smallest number with 12 divisors and this allows us to subdivide the hour equally in twelve different ways. 360 is the smallest number with 24 divisors and is the basis for our 360 degree system.
360 is a strange number anyway. The smallest number with 360 divisors is ... 3.603.600. How about that!
Even though integer multiplication seems as simple as can be. Integer division generates the strangest rhythmsand exceptions in the natural numbers. Creating graphs with strange jumps and fractal like abilities just by trying to divide numbers by others and wanting a whole number as an answer!